There are two additional relations, called indifference and strict preference that can be defined in terms of the preference relation . Both of these relations inherit
the properties of . In order to prove the existence of a utility function that represents choice, transitivity of the strict preference relation is needed.
Two consumption levels x and x' are indifferent if x x' and if x' x. This is typically written x x'.
Indifference is transitive because the relation that it is derived from is transitive.
Proof: If x x' and x' x'', then
from the definition of indifference above, x x'
and x' x'' so that x x'', and
also x'' x' and x' x so that
x'' x. Taken together, this shows that if
x x' and x' x'' then x x''.
Consumption level x is strictly preferred to consumption level x' if x x' and x' x. (The symbol mean "not".) In words, x is at least as good as x', but x' is not at least as good as x. This leaves only the possibility that x is strictly preferred to x', which we write as x x'.
Strict preference inherits transitivity from .
Proof: Let x and x' be such that x x' and x' x''. Then the strict preference relation is transitive if this implies that x x''.
From the definition of strict preference, x x' if and only if x x' and (x' x). Similarly, x' x'' if and only if x' x'' and (x'' x'). From x x' and x' x'' and transitivity of we get x x''. Suppose that x'' x. Since x x' this would imply, again by transitivity of , that x'' x'. But x' x'' which by definition implies that (x'' x'). This contradicts the assumption that x'' x, so x x'' and (x'' x), which is equivalent to x x''.